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20. Rheology & Linear Elasticity

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20. Rheology & Linear Elasticity 20. Rheology & Linear Elasticity I Main Topics A Rheology: Macroscopic deformation behavior B Linear elasticity for homogeneous isotropic materials 10/29/18 GG303 1 20. Rheology & Linear Elasticity Viscous (fluid) Behavior http://manoa.hawaii.edu/graduate/content/slide-lava 10/29/18 GG303 2 20. Rheology & Linear Elasticity Ductile (plastic) Behavior http://www.hilo.hawaii.edu/~csav/gallery/scientists/LavaHammerL.jpg http://hvo.wr.usgs.gov/kilauea/update/images.html 10/29/18 GG303 3 http://upload.wikimedia.org/wikipedia/commons/8/89/Ropy_pahoehoe.jpg 20. Rheology & Linear Elasticity Elastic Behavior https://thegeosphere.pbworks.com/w/page/24663884/Sumatra http://www.earth.ox.ac.uk/__Data/assets/image/0006/3021/seismic_hammer.jpg 10/29/18 GG303 4 20. Rheology & Linear Elasticity Brittle Behavior (fracture) 10/29/18 GG303 5 http://upload.wikimedia.org/wikipedia/commons/8/89/Ropy_pahoehoe.jpg 20. Rheology & Linear Elasticity II Rheology: Macroscopic deformation behavior A Elasticity 1 Deformation is reversible when load is removed 2 Stress (σ) is related to strain (ε) 3 Deformation is not time dependent if load is constant 4 Examples: Seismic (acoustic) waves, http://www.fordogtrainers.com rubber ball 10/29/18 GG303 6 20. Rheology & Linear Elasticity II Rheology: Macroscopic deformation behavior A Elasticity 1 Deformation is reversible when load is removed 2 Stress (σ) is related to strain (ε) 3 Deformation is not time dependent if load is constant 4 Examples: Seismic (acoustic) waves, rubber ball 10/29/18 GG303 7 20. Rheology & Linear Elasticity II Rheology: Macroscopic deformation behavior B Viscosity 1 Deformation is irreversible when load is removed 2 Stress (σ) is related to strain rate (ε ! ) 3 Deformation is time dependent if load is constant 4 Examples: Lava flows, corn syrup http://wholefoodrecipes.net 10/29/18 GG303 8 20. Rheology & Linear Elasticity II Rheology: Macroscopic deformation behavior B Viscosity 1 Deformation is irreversible when load is removed 2 Stress (σ) is related to strain rate (ε ! ) 3 Deformation is time dependent if load is constant 4 Examples: Lava flows, corn syrup 10/29/18 GG303 9 20. Rheology & Linear Elasticity II Rheology: Macroscopic deformation behavior C Plasticity 1 No deformation until yield strength is locally exceeded; then irreversible deformation occurs under a constant load 2 Deformation can increase with time under a constant load 3 Examples: plastics, soils http://www.therapyputty.com/images/stretch6.jpg 10/29/18 GG303 10 20. Rheology & Linear Elasticity II Rheology: Macroscopic deformation behavior C Brittle Deformation 1 Discontinuous deformation 2 Failure surfaces separate http://www.thefeeherytheory.com 10/29/18 GG303 11 20. Rheology & Linear Elasticity II Rheology: Macroscopic deformation behavior D Elasto-plastic rheology 10/29/18 GG303 12 20. Rheology & Linear Elasticity II Rheology: Macroscopic deformation behavior E Visco-plastic rheology 10/29/18 GG303 13 20. Rheology & Linear Elasticity II Rheology: Macroscopic deformation behavior F Power-law creep n (−Q/RT) 1 ε!= (σ1 − σ3) e 2 Example: rock salt 10/29/18 GG303 14 20. Rheology & Linear Elasticity II Rheology: Macroscopic deformation behavior G Linear vs. nonlinear behavior 10/29/18 GG303 15 20. Rheology & Linear Elasticity II Rheology: Macroscopic deformation behavior H Rheology=f (σij ,fluid pressure, strain rate, chemistry, temperature) I Rheologic equation of real rocks = ? 10/29/18 GG303 16 20. Rheology & Linear Elasticity II Rheology (cont.) J Experimental results Axial stress Plastic Elastic Increasing confining pressure Pc = confining pressure Compression test data on Tennessee marble II from Wawersik and Fairhurst, 1970 10/29/18 GG303 17 20. Rheology & Linear Elasticity III Linear elasticity x A Force and displacement F of a spring (from Hooke, 1676): F= kx 1 F = force 2 k = spring constant k Dimensions:F/L F 3 x = displacement Dimensions: length L) x 10/29/18 GG303 18 20. Rheology & Linear Elasticity σ III Linear elasticity (cont.) B Hooke’s Law for L0+ΔL L0 uniaxial stress: σ = Eε ε=ΔL/L0 1 σ = uniaxial stress 2 E = Young’s modulus Dimensions: stress E 3 ε = strain σ (elongation) Dimensionless ε 10/29/18 GG303 19 20. Rheology & Linear Elasticity Typical rock moduli and strengths Young’s Poisson’s Modulus Ratio (GPa) Uniaxial Strengths (MPa) Rock type Emin Emax νmin νmax Rock type Tensile Tensile Comp. Comp. (low) (high) (low) (high) Quartzite 70 105 0.11 0.25 Quartzite 17 28 200 304 Gneiss 16 103 0.10 0.40 Gneiss 3 21 73 340 Basalt 16 101 0.13 0.38 Basalt 2 28 42 355 Granite 10 74 0.10 0.39 Granite 3 39 30 324 Limestone 1 92 0.08 0.39 Limestone 2 40 48 210 Sandstone 10 46 0.10 0.40 Sandstone 3 7 40 179 Shale 10 44 0.10 0.19 Shale 2 5 36 172 Coal 1.5 3.7 0.33 0.37 Coal 1.9 3.2 14 30 10/29/18 GG303 20 20. Rheology & Linear Elasticity σ1 III Linear elasticity (cont.) B Hooke’s Law for uniaxial L +ΔL stress (cont.): ε1 = σ1/E 0 L0 ε=ΔL/L0 1 σ2 = σ3 = 0 2 ε2 = ε3 = -νε1 a ν = Poisson’s ratio b ν is diMensionless c Strain in one direction tends to induce strain in another direction 10/29/18 GG303 21 20. Rheology & Linear Elasticity σ1 III Linear elasticity (cont.) C Linear elasticity in 3D for L +ΔL 0 L0 homogeneous isotropic ε=ΔL/L0 materials By superposition: 1 εxx = σxx/E – (σyy+σzz)(ν/E) 2 εyy = σyy/E – (σzz+σxx)(ν/E) 3 εzz = σzz/E – (σxx+σyy)(ν/E) 10/29/18 GG303 22 20. Rheology & Linear Elasticity σ1 III Linear elasticity (cont.) C Linear elasticity in 3D for homogeneous isotropic L0+ΔL L0 ε=ΔL/L materials (cont.) 0 4 Directions of principal stresses and principal strains coincide 5 Extension in one direction can occur without tension 6 Compression in one direction can occur without shortening 10/29/18 GG303 23 20. Rheology & Linear Elasticity III Linear elasticity E Special cases 1 Isotropic (hydrostatic) stress a σ1 = σ2 = σ3 b No shear stress 2 Uniaxial strain x a εxx= ε1≠ 0 b εyy = εzz = 0 10/29/18 GG303 24 20. Rheology & Linear Elasticity III Linear elasticity E Special cases 3 Plane stress (2D) σz = 0 “Thin plate” case 4 Plane strain (2D) εz = 0 a Displacement in z- direction is constant (e.g., zero) b Plate is confined between rigid walls c “Thick plate” case 10/29/18 GG303 25 20. Rheology & Linear Elasticity III Linear elasticity E Special cases 5 Pure shear stress (2D) σxx= -σyy; σzz=0 10/29/18 GG303 26 20. Rheology & Linear Elasticity III Linear elasticity F Strain energy (W0) for uniaxial stress σxxdydz 1 2 W = (1/2)(σxxdydz) (εxxdx) ε dx 3 W = (1/2) (σxxεxx) (dxdydz) xx 4 W0 = W/(dxdydz) W0 = strain energy density 5 W0 = (1/2)(σxxεxx) 10/29/18 GG303 27 20. Rheology & Linear Elasticity IIILinear elasticity G Strain energy (W0) in 3D σxxdydz W0 = (1/2)(σ1ε1+σ2ε2+σ3ε3) εxxdx 10/29/18 GG303 28 20. Rheology & Linear Elasticity III Linear elasticity 4 β = compressibility D Relationships among β = 1/K different elastic moduli ∆ = εxx+εyy+ εzz= -p/K 1 G = μ = shear modulus p = pressure G = E/(2[1+ν]) 5 P-wave speed: Vp ε = σ /2G xy xy ⎛ 4 ⎞ 2 λ = Lame' constant Vp = K + µ ρ ⎝⎜ 3 ⎠⎟ λ = Ev/([1 + ν][1 - 2ν]) 6 S-wave speed: Vs 3 K = bulk modulus K = E/(3[1 - 2ν]) 10/29/18 GG303 29.
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